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Cosmological Solutions, a New Wick-Rotation, and the First Law of Thermodynamics
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Abstract
We present a modified implementation of the Euclidean action formalism suitable for studying the thermodynamics of a class of cosmological solutions containing Killing horizons. To obtain a real metric of definite signature, we perform a `triple Wick-rotation' by analytically continuing all space-like directions. The resulting Euclidean geometry is used to calculate the Euclidean on-shell action, which defines a thermodynamic potential. We show that for the vacuum de Sitter solution, planar solutions of Einstein-Maxwell theory and a previously found class of cosmological solutions of N = 2 supergravity, this thermodynamic potential can be used to define an internal energy which obeys the first law of thermodynamics. Our approach is complementary to, but consistent with the isolated horizon formalism. For planar Einstein-Maxwell solutions, we find dual solutions in Einstein-anti-Maxwell theory where the sign of the Maxwell term is reversed. These solutions are planar black holes, rather than cosmological solutions, but give rise, upon a standard Wick-rotation to the same Euclidean action and thermodynamic relations.
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A bstract
Spinorial geometry methods are used to classify solutions admitting Majorana Killing spinors of the minimal 4-dimensional supergravity in neutral signature, with van- ishing cosmological constant and a single Maxwell field strength. Two classes of solutions preserving the minimal amount of supersymmetry are found. The first class admits a null- K¨ahler structure and corresponds to a class of self-dual solutions found by Bryant. The second class admits a null and rotation-free geodesic congruence with respect to which a parallel frame can be chosen. Examples of solutions in the former class are pseudo-hyper- K¨ahler manifolds; and examples in the latter class include self-dual solutions, as well as a neutral-signature IWP-type solution.
A bstract
We obtain cosmological solutions with Kasner-like asymptotics in $$ \mathcal{N} $$ N = 2 gauged and ungauged supergravity by maximal analytic continuation of planar versions of non-extremal black hole solutions. Initially, we construct static solutions with planar symmetry by solving the time-reduced field equations. Upon lifting back to four dimensions, the resulting static regions are incomplete and bounded by a curvature singularity on one side and a Killing horizon on the other. Analytic continuation reveals the existence of dynamic patches in the past and future, with Kasner-like asymptotics. For the ungauged STU-model, our solutions contain previously known solutions with the same conformal diagram as a subset. We find explicit lifts to five, six, ten and eleven dimensions which show that in the extremal limit, the underlying brane configuration is the same as for STU black holes. The extremal limit of the six-dimensional lift is shown to be BPS for special choices of the integration constants. We argue that there is a universal correspondence between spherically symmetric black hole solutions and planar cosmological solutions which can be illustrated using the Reissner-Nordström solution of Einstein-Maxwell theory.
A bstract
We start developing a formalism which allows to construct supersymmetric theories systematically across space-time signatures. Our construction uses a complex form of the supersymmetry algebra, which is obtained by doubling the spinor representation. This allows one to partially disentangle the Lorentz and R-symmetry group and generalizes symplectic Majorana spinors. For the case where the spinor representation is complex-irreducible, the R-symmetry only acts on an internal multiplicity space, and we show that the connected groups which occur are SO(2), SO 0 (1, 1), SU(2) and SU(1, 1).
As an application we construct the off-shell supersymmetry transformations and supersymmetric Lagrangians for five-dimensional vector multiplets in arbitrary signature ( t , s ). In this case the R-symmetry groups are SU(2) or SU(1, 1), depending on whether the spinor representation carries a quaternionic or para-quaternionic structure. In Euclidean signature the scalar and vector kinetic terms differ by a relative sign, which is consistent with previous results in the literature and shows that this sign flip is an inevitable consequence of the Euclidean supersymmetry algebra.
We construct N=2 four and five-dimensional supergravity theories coupled to vector multiplets in various space-time signatures (t,s), where t and s refer, respectively, to the number of time and spatial dimensions. The five-dimensional supergravity theories, t+s=5, are constructed by investigating the integrability conditions arising from Killing spinor equations. The five-dimensional supergravity theories can also be obtained by reducing Hull's eleven-dimensional supergravities on a Calabi-Yau threefold. The dimensional reductions of the five-dimensional supergravities on space and time-like circles produce N=2 four-dimensional supergravity theories with signatures (t-1,s) and (t,s-1) exhibiting projective special (para)-K\"ahler geometry.
We discuss phantom metrics admitting Killing spinors in fake N=2, D=4
supergravity coupled to vector multiplets. The Abelian U(1) gauge fields in the
fake theory have kinetic terms with the wrong sign. We solve the Killing spinor
equations for the standard and fake theories in a unified fashion by
introducing a parameter which distinguishes between the two theories. The
solutions found are fully determined in terms of algebraic conditions, the
so-called stabilisation equations, in which the symplectic sections are related
to a set of functions. These functions are harmonic in the case of the standard
supergravity theory and satisfy the wave-equation in flat (2+1)-space-time in
the fake theory. Explicit examples are given for the minimal models with
quadratic prepotentials.
We study metric solutions of Einstein-anti-Maxwell theory admitting Killing
spinors. The analogue of the IWP metric which admits a space-like Killing
vector is found and is expressed in terms of a complex function satisfying the
wave equation in flat (2+1)-dimensional space-time. As examples, electric and
magnetic Kasner spaces are constructed by allowing the solution to depend only
on the time coordinate. Euclidean solutions are also presented.
We work with the notion of apparent/trapping horizons for spherically
symmetric, dynamical spacetimes: these are quasi-locally defined, simply based
on the behaviour of congruence of light rays. We show that the sign of the
dynamical Hayward-Kodama surface gravity is dictated by the inner/outer nature
of the horizon. Using the tunneling method to compute Hawking Radiation, this
surface gravity is then linked to a notion of temperature, up to a sign that is
dictated by the future/past nature of the horizon. Therefore two sign effects
are conspiring to give a positive temperature for the black hole case and the
expanding cosmology, whereas the same quantity is negative for white holes and
contracting cosmologies. This is consistent with the fact that, in the latter
cases, the horizon does not act as a separating membrane, and Hawking emission
should not occur.
We classify the supersymmetric solutions of minimal $N=2$ gauged supergravity
in four dimensions with neutral signature. They are distinguished according to
the sign of the cosmological constant and whether the vector field constructed
as a bilinear of the Killing spinor is null or non-null. In neutral signature
the bilinear vector field can be spacelike, which is a new feature not arising
in Lorentzian signature. In the $\Lambda<0$ non-null case, the canonical form
of the metric is described by a fibration over a three-dimensional base space
that has $\text{U}(1)$ holonomy with torsion. We find that a generalized
monopole equation determines the twist of the bilinear Killing field, which is
reminiscent of an Einstein-Weyl structure. If, moreover, the electromagnetic
field strength is self-dual, one gets the Kleinian signature analogue of the
Przanowski-Tod class of metrics, namely a pseudo-hermitian spacetime determined
by solutions of the continuous Toda equation, conformal to a scalar-flat
pseudo-K\"ahler manifold, and admitting in addition a charged conformal Killing
spinor. In the $\Lambda<0$ null case, the supersymmetric solutions define an
integrable null K\"ahler structure. In the $\Lambda>0$ non-null case, the
manifold is a fibration over a Lorentzian Gauduchon-Tod base space. Finally, in
the $\Lambda>0$ null class, the metric is contained in the Kundt family, and it
turns out that the holonomy is reduced to ${\rm Sim}(1)\times{\rm Sim}(1)$.
There appear no self-dual solutions in the null class for either sign of the
cosmological constant.
We construct new black brane solutions in $U(1)$ gauged ${\cal N}=2$
supergravity with a general cubic prepotential, which have entropy density
$s\sim T^{1/3}$ as $T \rightarrow 0$ and thus satisfy the Nernst Law. By using
the real formulation of special geometry, we are able to obtain analytical
solutions in closed form as functions of two parameters, the temperature $T$
and the chemical potential $\mu$. Our solutions interpolate between
hyperscaling violating Lifshitz geometries with $(z,\theta)=(0,2)$ at the
horizon and $(z,\theta)=(1,-1)$ at infinity. In the zero temperature limit,
where the entropy density goes to zero, we recover the extremal Nernst branes
of Barisch et al, and the parameters of the near horizon geometry change to
$(z,\theta)=(3,1)$.
We construct new static, spherically symmetric non-extremal black hole solutions of four-dimensional \( \mathcal{N}=2 \) supergravity, using a systematic technique based on dimensional reduction over time (the c-map) and the real formulation of special geometry. For a certain class of models we actually obtain the general solution to the full second order equations of motion, whilst for other classes of models, such as those obtainable by dimensional reduction from five dimensions, heterotic tree-level models, and type-II Calabi-Yau compactifications in the large volume limit a partial set of solutions are found.
When considering specifically non-extremal black hole solutions we find that regularity conditions reduce the number of integration constants by one half. Such solutions satisfy a unique set of first order equations, which we identify.
Several models are investigated in detail, including examples of non-homogeneous spaces such as the quantum deformed STU model. Though we focus on static, spherically symmetric solutions of ungauged supergravity, the method is adaptable to other types of solutions and to gauged supergravity.
We exploit the parallel between dynamical black holes and cosmological
spacetimes to describe the evolution of Friedmann-Lema\^itre-Robertson-Walker
universes from the point of view of an observer in terms of the dynamics of the
apparent horizon. Using the Hayward-Kodama formalism of dynamical black holes,
we clarify the role of the Clausius relation to derive the Friedmann equations
for a universe, in the spirit of Jacobson's work on the thermodynamics of
spacetime. We also show how dynamics at the horizon naturally leads to the
quantum-mechanical process of Hawking radiation. We comment on the connection
of this work with recent ideas to consider our observable Universe as a
Bose-Einstein condensate and on the corresponding role of vacuum energy.
A framework was recently introduced to generalize black hole mechanics by replacing stationary event horizons with isolated horizons. That framework is significantly extended. The extension is non-trivial in that not only do the boundary conditions now allow the horizon to be distorted and rotating, but also the subsequent analysis is based on several new ingredients. Specifically, although the overall strategy is closely related to that in the previous work, the dynamical variables, the action principle and the Hamiltonian framework are all quite different. More importantly, in the non-rotating case, the first law is shown to arise as a necessary and sufficient condition for the existence of a consistent Hamiltonian evolution. Somewhat surprisingly, this consistency condition in turn leads to new predictions even for static black holes. To complement the previous work, the entire discussion is presented in terms of tetrads and associated (real) Lorentz connections. Comment: 56 pages, 1 figure, Revtex; Final Version, to appear in PRD
The target space geometry of abelian vector multiplets in N=2 theories in four and five space–time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometries of four- and five-dimensional vector multiplets. We formulate N=2 theories in terms of Hessian structures and give various concrete applications of Hessian geometry, ranging from static BPS black holes in four and five space–time dimensions to topological string theory, emphasizing the role of the Hesse potential. We also discuss the r-map and c-map which relate the special geometries of vector multiplets to each other and to hypermultiplet geometries. By including time-like dimensional reductions, we obtain theories in Euclidean signature, where the scalar target spaces carry para-complex versions of special geometry.
Recently, definitions of total 4-momentum and angular momentum of isolated gravitating systems have been introduced in terms of the asymptotic behavior of the Weyl curvature (of the underlying space-time) at spatial infinity. Given a space-time equipped with isometries, on the other hand, one can also construct conserved quantities using the presence of the Killing fields. Thus, for example, for stationary space-times, the Komar integral can be used to define the total mass, and, the asymptotic value of the twist of the Killing field, to introduce the dipole angular momentum moment. Similarly, for axisymmetric space-times, one can obtain the ("z-component" of the) total angular momentum in terms of the Komar integral. It is shown that, in spite of their apparently distinct origin, in the presence of isometries, quantities defined at spatial infinity reduce to the ones constructed from Killing fields. This agreement reflects one of the many subtle aspects of Einstein's (vacuum) equation.
We construct Nernst brane solutions, that is black branes with zero entropy density in the extremal limit, of FI-gauged minimal five-dimensional supergravity coupled to an arbitrary number of vector multiplets. While the scalars take specific constant values and dynamically determine the value of the cosmological constant in terms of the FI-parameters, the metric takes the form of a boosted AdS Schwarzschild black brane. This metric can be brought to the Carter-Novotny-Horsky form that has previously been observed to occur in certain limits of boosted D3-branes. By dimensional reduction to four dimensions we recover the four-dimensional Nernst branes of arXiv:1501.07863 and show how the five-dimensional lift resolves all their UV singularities. The dynamics of the compactification circle, which expands both in the UV and in the IR, plays a crucial role. At asymptotic infinity, the curvature singularity of the four-dimensional metric and the run-away behaviour of the four-dimensional scalar combine in such a way that the lifted solution becomes asymptotic to AdS5. Moreover, the existence of a finite chemical potential in four dimensions is related to fact that the compactification circle has a finite minimal value. While it is not clear immediately how to embed our solutions into string theory, we argue that the same type of dictionary as proposed for boosted D3-branes should apply, although with a lower amount of supersymmetry.
There are a number of similarities between black-hole physics and thermodynamics. Most striking is the similarity in the behaviors of black-hole area and of entropy: Both quantities tend to increase irreversibly. In this paper we make this similarity the basis of a thermodynamic approach to black-hole physics. After a brief review of the elements of the theory of information, we discuss black-hole physics from the point of view of information theory. We show that it is natural to introduce the concept of black-hole entropy as the measure of information about a black-hole interior which is inaccessible to an exterior observer. Considerations of simplicity and consistency, and dimensional arguments indicate that the black-hole entropy is equal to the ratio of the black-hole area to the square of the Planck length times a dimensionless constant of order unity. A different approach making use of the specific properties of Kerr black holes and of concepts from information theory leads to the same conclusion, and suggests a definite value for the constant. The physical content of the concept of black-hole entropy derives from the following generalized version of the second law: When common entropy goes down a black hole, the common entropy in the black-hole exterior plus the black-hole entropy never decreases. The validity of this version of the second law is supported by an argument from information theory as well as by several examples.
The unconstrained dynamical degrees of freedom of the gravitational field are identified with the conformally invariant three-geometries of spacelike hypersurfaces. New results concerning the action principle, choice of canonical variables, and initial-value equations strengthen this identification. One of the new canonical variables is shown to play the role of "time" in the formalism.
The problem of the dynamical structure and definition of energy for the classical general theory of relativity is considered on a formal level. As in a previous paper, the technique used is the Schwinger action principle. Starting with the full Einstein Lagrangian in first order Palatini form, an action integral is derived in which the algebraic constraint variables have been eliminated. This action possesses a "Hamiltonian" density which, however, vanishes due to the differential constraints. If the differential constraints are then substituted into the action, the true, nonvanishing Hamiltonian of the theory emerges. From an analysis of the equations of motion and the constraint equations, the two pairs of dynamical variables which represent the two independent degrees of freedom of the gravitational field are explicitly exhibited. Four other variables remain in theory; these may be arbitrarily specified, any such specification representing a choice of coordinate frame. It is shown that it is possible to obtain truly canonical pairs of variables in terms of the dynamical and arbitrary variables. Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field. There thus arises the notion of a preferred coordinate frame, i.e., that frame in which the Hamiltonian is conserved. In this special frame, on physical grounds, the Hamiltonian may be taken to define the energy of the field. In these respects the situation in general relativity is analogous to the parametric form of Hamilton's principle in particle mechanics.
The circumstances under which negative absolute temperatures can occur are discussed, and principles of thermodynamics and statistical mechanics at negative temperatures are developed. If the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever (∂S/∂U)X is negative. Negative temperatures are hotter than positive temperatures. When account is taken of the possibility of negative temperatures, various modifications of conventional thermodynamics statements are required. For example, heat can be extracted from a negative-temperature reservoir with no other effect than the performance of an equivalent amount of work. One of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and provide no effect other than (1) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated, and that has an energetic upper limit to its allowed states can possess a negative temperature. The statistical mechanics of such a system are discussed and the results are applied to nuclear spin systems.
For the spherically symmetric system, we prove the existence of a new locally conserved flux which can be interpreted to represent the total energy flux of matter and gravitational field. With the aid of this conservation law, we study the relation between the behavior of the event horizon and the energy flux across it and look for constraints imposed on the total energy radiated to infinity. Some implications of the results of this study to the backreaction problem in the black hole evaporation are discussed.
It is shown that the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant. An observer in these models will have an event horizon whose area can be interpreted as the entropy or lack of information of the observer about the regions which he cannot see. Associated with the event horizon is a surface gravity κ which enters a classical "first law of event horizons" in a manner similar to that in which temperature occurs in the first law of thermodynamics. It is shown that this similarity is more than an analogy: An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon. If the observer absorbs some of this radiation, he will gain energy and entropy at the expense of the region beyond his ken and the event horizon will shrink. The derivation of these results involves abandoning the idea that particles should be defined in an observer-independent manner. They also suggest that one has to use something like the Everett-Wheeler interpretation of quantum mechanics because the back reaction and hence the spacetime metric itself appear to be observer-dependent, if one assumes, as seems reasonable, that the detection of a particle is accompanied by a change in the gravitational field.
One can evaluate the action for a gravitational field on a section of the complexified spacetime which avoids the singularities. In this manner we obtain finite, purely imaginary values for the actions of the Kerr-Newman solutions and de Sitter space. One interpretation of these values is that they give the probabilities for finding such metrics in the vacuum state. Another interpretation is that they give the contribution of that metric to the partition function for a grand canonical ensemble at a certain temperature, angular momentum, and charge. We use this approach to evaluate the entropy of these metrics and find that it is always equal to one quarter the area of the event horizon in fundamental units. This agrees with previous derivations by completely different methods. In the case of a stationary system such as a star with no event horizon, the gravitational field has no entropy.
I address the question: What is fixed on the boundary in the action principles of general relativity? Four forms of the action are considered: the Einstein action, the Hilbert action, the first order action, and what may be called the cosmological action. The relationships and boundary data of these actions are described geometrically. Formal passage to the Euclidean forms of these actions is effected in detail.
In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects
cause black holes to create and emit particles as if they were hot bodies with temperature
\frachk2pk » 10 - 6 ( \fracM\odot M )° K\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K
where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black
hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon
of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons
in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so
much entropy per baryon.
Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the ``surface gravity'' kappa of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.
These notes are loosely based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge theories, February 2009 and at the IPM String School in Tehran, April 2009. I have focused on a few concrete topics and also on addressing questions that have arisen repeatedly. Background condensed matter physics material is included as motivation and easy reference for the high energy physics community. The discussion of holographic techniques progresses from equilibrium, to transport and to superconductivity. Comment: 1+85 pages. 15 figures. v2: typos fixed and references added. v3: another typo fixed
A number of attempts have recently been made to extend the conjectured $S$ duality of Yang Mills theory to gravity. Central to these speculations has been the belief that electrically and magnetically charged black holes, the solitons of quantum gravity, have identical quantum properties. This is not obvious, because although duality is a symmetry of the classical equations of motion, it changes the sign of the Maxwell action. Nevertheless, we show that the chemical potential and charge projection that one has to introduce for electric but not magnetic black holes exactly compensate for the difference in action in the semi-classical approximation. In particular, we show that the pair production of electric black holes is not a runaway process, as one might think if one just went by the action of the relevant instanton. We also comment on the definition of the entropy in cosmological situations, and show that we need to be more careful when defining the entropy than we are in an asymptotically-flat case.
We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is not precisely thermal. We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor.
Versions of M-theory are found in spacetime signatures (9,2) and (6,5), in addition to the usual M-theory in 10+1 dimensions, and these give rise to type IIA string theories in 10-dimensional spacetime signatures (10,0),(9,1),(8,2),(6,4) and (5,5), and to type IIB string theories in signatures (9,1),(7,3) and (5,5). The field theory limits are 10 and 11 dimensional supergravities in these signatures. These theories are all linked by duality transformations which can change the number of time dimensions as well as the number of space dimensions, so that each should be a different limit of the same underlying theory. Comment: Minor changes. 52 pages, 14 Postscript figures, uses phyzzx macro
T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type $IIB^*$ theory and the IIB theory to a type $IIA^*$ theory. The type $II^*$ theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an E$n$-brane is the $n$-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on $9-n$ spatial dimensions and one time dimension. The $IIB^*$ theory has a solution which is the product of 5-dimensional de Sitter space and a 5-hyperboloid, and the E4-brane corresponds to anon-singular complete solution which interpolates between this solution and flat space. This leads to a duality between the large $N$ limit of the Euclidean 4-dimensional U(N) super-Yang-Mills theory and the $IIB^*$ string theory in de Sitter space, and both are invariant under the same de Sitter supergroup. This theory can be twisted to obtain a large $N$ topological gauge theory and its topological string theory dual. Flat space-time may be an unstable vacuum for the type $II^*$ theories, but they have supersymmetric cosmological solutions. Comment: 47 pages, 1 Postscript figure, uses phyzzx macro. Monor corrections
These lectures present an elementary discussion of some background material relevant to the problem of de Sitter quantum gravity. The first two lectures discuss the classical geometry of de Sitter space and properties of quantum field theory on de Sitter space, especially the temperature and entropy of de Sitter space. The final lecture contains a pedagogical discussion of the appearance of the conformal group as an asymptotic symmetry group, which is central to the dS/CFT correspondence. A (previously lacking) derivation of asymptotically de Sitter boundary conditions is also given.
A unified first law of black-hole dynamics and relativistic thermodynamics is derived in spherically symmetric general relativity. This equation expresses the gradient of the active gravitational energy E according to the Einstein equation, divided into energy-supply and work terms. Projecting the equation along the flow of thermodynamic matter and along the trapping horizon of a blackhole yield, respectively, first laws of relativistic thermodynamics and black-hole dynamics. In the black-hole case, this first law has the same form as the first law of black-hole statics, with static perturbations replaced by the derivative along the horizon. There is the expected term involving the area and surface gravity, where the dynamic surface gravity is defined as in the static case but using the Kodama vector and trapping horizon. This surface gravity vanishes for degenerate trapping horizons and satisfies certain expected inequalities involving the area and energy. In the thermodynamic case, the quasi-local first law has the same form, apart from a relativistic factor, as the classical first law of thermodynamics, involving heat supply and hydrodynamic work, but with E replacing the internal energy. Expanding E in the Newtonian limit shows that it incorporates the Newtonian mass, kinetic energy, gravitational potential energy and thermal energy. There is also a weak type of unified zeroth law: a Gibbs-like definition of thermal equilibrium requires constancy of an effective temperature, generalising the Tolman condition and the particular case of Hawking radiation, while gravithermal equilibrium further requires constancy of surface gravity. Finally, it is suggested that the energy operator of spherically symmetric quantum gravity is determined by the Kodama vector, which encodes a dynamic time related to E.
The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on ${}^3B$, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate ${}^3B$. The integral of the energy density over such a two-surface $B$ is the quasilocal energy associated with a spacelike three-surface $\Sigma$ whose intersection with ${}^3B$ is the boundary $B$. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary $B$ as a surface embedded in $\Sigma$. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on ${}^3B$ in the direction orthogonal to $B$. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on ${}^3B$. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.
Black hole entropy, special geometry and strings
- T Mohaupt
T. Mohaupt, Black hole entropy, special geometry and strings,
Fortsch. Phys. 49 (2001) 3 [hep-th/0007195].
Kasner Branes with Arbitrary Signature
- W Sabra
W. Sabra, Kasner Branes with Arbitrary Signature, 2005.03953.
Exact Space-Times in Einstein's General Relativity
- J B Griffiths
- J Podolsky
J. B. Griffiths and J. Podolsky, Exact Space-Times in Einstein's
General Relativity. Cambridge University Press, Cambridge, 2009.